Dependence of composition on the potential

We first consider a closed, single-phase system containing C components in a gravitational field. The state of the system is fixed and the system is in equilibrium. Equation (14.25) is applicable to each component in the system. We have then a set of equations, C in number, relating (8P/8r)T and the (C — 1) derivatives (dXj/dr)T. The solution of this set of equations then gives equations for each of the derivatives. Equation (14.26) might be used to replace (8P/8r)T, but this equation is not independent of the set of equations given by Equation (14.25) because it is derived from this set. If Equation (14.26) is used, then one of the C equations from Equation (14.25) would be omitted. 

The complete solution of the problem would require the integration of the derivatives to obtain P and each independent x, as a function of r, with the knowledge of the required integration constants. These would be obtained by fixing the value of the pressure at some position in the field and also fixing the value of each mole fraction at some position in the field. 

As an illustration let us consider a two-component system, make use of Equation (14.26), and use Xj as the independent mole fraction. The set of equations are 

This set may be solved in principle to yield P and Xj, each as a function of r, with the knowledge of a value of P and Xj for fixed values of r. The solution is easily obtained on the assumption that the ideal solution laws are applicable and that either the ideal gas equation is followed in the case of a gas phase or the volumes are independent of the pressure. Then, for an ideal liquid solution, Equation (14.27) becomes 

Similar calculations would be carried out for heterogenous systems provided the phase boundary was wholly confined in a single region. The pressure at the phase boundary would be the same on either side of the boundary, as described in Section 14.4. The composition on one side of the boundary would be determined from knowledge of the composition on the 

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